Until recently,
based on the seminal perturbative calculation of Luttinger [1]},
it was generally assumed that the Luttinger
integrals \(I_{\tau \sigma }\)  vanish (at least for \(B=0\) ).
However, several cases are now known where \(I_{\tau \sigma }=\pm \pi /2\)  [2]}, [3]}, [4]}, [5]}, [6]},
yet the quasiparticle scattering phase shifts show no
low-energy singularities.
In particular in our case for \(B=0\) 
and both orbitals equivalent, the four \(I_{\tau \sigma }=I_{L}\)  are equal
by symmetry and \(I_{L}\)  has a topological character, being equal
to 0 for \(D<D_{c}\)  and \(\pi /2\)  for \(D>D_{c}\)  [3]}, [4]}.
In this case,
Eqs. (REF ) and (REF ) imply that in the Kondo limit,
where \(\left\langle n_{\tau \sigma }\right\rangle =1/2\) , the spectral densities \(\rho _{\tau \sigma }(0)\)  jump
from \(\rho _{0}=1/(\pi \Delta _{\tau })\)  for \(D<D_{c}\)  to 0 for \(D>D_{c}\) .
In the general case, when the channels are not equivalent and a magnetic field is present,
we show in Supplementary Note 2
that the conservation laws imply that
three topological quantities \(T,\)  \(T_{\tau },\)  \(T_{\sigma }\)  can still be
defined:
\(T=\sum _{\tau \sigma }I_{_{\tau \sigma }},\text{ }T_{\tau }=\sum _{\tau \sigma }\tau I_{_{\tau \sigma }},\text{ }T_{\sigma }=\sum _{\tau \sigma }\sigma I_{_{\tau \sigma }}, \) 
